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High-dimensional quantum process tomography with undetected photons

This paper proposes an interferometric method for high-dimensional quantum process tomography that fully reconstructs arbitrary operations on qudits without requiring any direct measurements on the transformed quantum states.

Original authors: Salini Rajeev, Mayukh Lahiri

Published 2026-08-04

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CC BY 4.0

Original authors: Salini Rajeev, Mayukh Lahiri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out exactly how a mysterious machine changes the things you put inside it. In the world of quantum physics, this machine is called a "quantum process," and the things it changes are tiny particles of light called photons. To understand how these machines work, scientists usually have to catch the particle after it exits the machine and measure it. But here's the catch: sometimes the particle changes into a color or type of light that our eyes and cameras can't see, or the detectors we have just aren't good enough to catch it. It's like trying to figure out how a magic trick works by only looking at the magician's hands, but the real trick happens in a room where you aren't allowed to look. This is a big problem because scientists want to build super-fast, super-secure communication networks using these high-dimensional quantum states, but they can't test them properly if they can't see the results.

This is where a new idea comes in, using a clever trick called "path identity." Think of it like a game of "telephone" played with light. Instead of looking at the particle that went through the mystery machine, scientists create a twin particle that never goes through the machine at all. By making the paths of these twins so perfectly matched that you can't tell which is which, the "untouched" twin starts to carry the secret fingerprints of what happened to its partner. This allows scientists to reconstruct the entire operation of the machine just by looking at the twin that stayed safe and sound, without ever needing to catch the one that went through the unknown process.

The paper by Salini Rajeev and Mayukh Lahiri takes this concept and pushes it further than ever before. While previous experiments used this "twin" trick only for machines that simply shuffle things around without losing any energy (called "unitary" operations), the authors show that it works for messy, real-world machines too. In the real world, machines often lose energy or change things in ways that aren't perfect; these are called "non-unitary" operations. The team proves that even when the machine is imperfect and acts like a leaky bucket rather than a perfect shuffle, you can still fully map out exactly what it does. They demonstrate this using a special kind of light called "orbital angular momentum" (which you can imagine as light twisting like a corkscrew), showing that by measuring only the safe, untouched twin photons, they can mathematically rebuild the entire blueprint of the mysterious machine.

The researchers didn't just guess this would work; they built a detailed mathematical model using quantum field theory to prove it. They showed that by sending a known, controlled twist (a "unitary transformation") onto the safe twin and then watching how the two light beams interfere with each other, they can read off the exact settings of the unknown machine. They tested their theory with a specific example of a non-unitary machine that had losses built into it, and their method successfully retrieved all the hidden numbers describing how that machine worked. The paper confirms that this "undetected photon" method is a powerful new tool. It means that in the future, scientists might be able to test and verify complex quantum devices even if the light coming out of them is in a wavelength range where no detectors exist yet, opening the door to exploring new parts of the light spectrum for quantum technology.

Technical Summary: High-dimensional Quantum Process Tomography with Undetected Photons

Problem Statement
Quantum process tomography (QPT) is essential for the complete characterization of quantum operations, a prerequisite for benchmarking and validating quantum networks. While high-dimensional quantum systems (qudits), particularly those encoded in photonic states like orbital angular momentum (OAM), offer superior information capacity and noise tolerance compared to qubits, characterizing them presents significant challenges. Existing QPT methods, including ancilla-assisted techniques, fundamentally rely on detecting the photon that has undergone the transformation. This requirement limits the exploration of specific spectral regions, such as the mid- and far-infrared, where adequate single-photon detectors are unavailable. Furthermore, while recent advances have demonstrated the reconstruction of high-dimensional unitary transformations without detecting the transformed qudit using path identity, these methods do not extend to non-unitary operations, which are common in practical scenarios and involve loss or decoherence.

Methodology
The authors propose an interferometric scheme based on the Zou-Wang-Mandel interferometer to perform QPT on arbitrary linear quantum operations (both unitary and non-unitary) without detecting the transformed photon. The setup utilizes two mutually coherent, identical sources ( $Q_1$ and $Q_2$ ) that generate high-dimensional Bell states consisting of signal ( $S$ ) and idler ( $I$ ) photon pairs.

Path Identity and Coherence: The idler photon from source $Q_1$ is sent through source $Q_2$ and perfectly aligned with the idler photon generated by $Q_2$ . This "path identity" erases "which-path" information for the signal photons, inducing coherence between the signal beams $S_1$ and $S_2$ .
Undetected Transformation: The unknown quantum operation $\hat{T}$ (potentially non-unitary) is applied to the idler photon in beam $I_1$ . Crucially, this idler photon is never detected.
Controlled Unitary and Interference: A known, controllable unitary transformation $\hat{U}$ is applied to the signal photon in beam $S_1$ . The signal beams $S_1$ and $S_2$ are then superposed at a beamsplitter, and a single-photon detector measures the output after projecting onto a specific OAM mode.
Theoretical Framework: The authors address the theoretical challenge of non-unitary transformations, which do not preserve field commutation relations. They introduce a vacuum field term to analytically treat non-unitary operators, ensuring the transformed field remains a legitimate quantum operator. This formalism unifies the treatment of both unitary and non-unitary operations within the same interferometric framework.

Key Results and Reconstruction Protocol
The method relies on analyzing the single-photon interference pattern observed at the detector. The detection probability $P_d$ exhibits sinusoidal variation dependent on the interferometric phase, with the visibility and phase of the pattern encoding the matrix elements of the unknown operator $\hat{T}$ .

Diagonal Elements: By setting the known unitary $\hat{U}$ to the identity matrix, the interference pattern reveals the magnitude and argument of the diagonal elements of $\hat{T}$ .
Off-Diagonal Elements: To retrieve arbitrary matrix elements, the authors propose applying specific two-dimensional rotations in the OAM space to the signal beam. By varying the rotation angle $\theta$ (specifically between $0$ and $\pi/2$ ) and detecting specific modes, the visibility of the resulting interference patterns yields the magnitudes of the matrix elements, while the phase shifts provide their arguments.
Efficiency: To fully reconstruct an $N$ -dimensional unknown operation, the protocol requires $N(N-1)/2$ choices of the rotation matrix $\hat{U}_b(q,r)$ .
Demonstration: The authors provide a theoretical numerical example demonstrating the reconstruction of a non-unitary $4 \times 4$ matrix derived from a Hadamard gate with mode-dependent losses. By applying six specific rotation settings, they show that all 16 matrix elements, including complex values, can be retrieved solely from the interference patterns of the signal photons.

Significance and Claims
The paper claims to demonstrate that quantum process tomography can be performed on high-dimensional quantum states without ever detecting the photon that interacted with the unknown process. The primary significance lies in extending the path-identity technique, previously limited to unitary transformations, to the broader class of linear non-unitary operations.

The authors emphasize that this approach overcomes the detector limitations associated with specific wavelength regions, as the detection is performed on the signal photon (which can be chosen to be in a detectable range, e.g., visible or near-infrared) while the transformation occurs on the idler photon (which could be in a hard-to-detect range). The method is grounded in quantum field theory, suggesting its potential applicability to non-photonic quantum systems in principle. The work establishes a theoretical and experimental pathway for characterizing quantum processes in regimes where direct detection of the transformed state is currently impossible.

Technical Summary: High-dimensional Quantum Process Tomography with Undetected Photons

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